Calculus Examples

$\int_{0}^{\frac{π}{2}} \sin^{7} (x) \cos^{5} (x) d x$

Step 1

Factor out $\cos^{4} (x)$ .

$\int_{0}^{\frac{π}{2}} \sin^{7} (x) (\cos^{4} (x) \cos (x)) d x$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $2$ out of $4$ .

$\int_{0}^{\frac{π}{2}} \sin^{7} (x) ({\cos (x)}^{2 (2)} \cos (x)) d x$

Step 2.2

Rewrite ${\cos (x)}^{2 (2)}$ as exponentiation.

$\int_{0}^{\frac{π}{2}} \sin^{7} (x) ({(\cos^{2} (x))}^{2} \cos (x)) d x$

$\int_{0}^{\frac{π}{2}} \sin^{7} (x) ({(\cos^{2} (x))}^{2} \cos (x)) d x$

Step 3

Using the Pythagorean Identity, rewrite $\cos^{2} (x)$ as $1 - \sin^{2} (x)$ .

$\int_{0}^{\frac{π}{2}} \sin^{7} (x) ({(1 - \sin^{2} (x))}^{2} \cos (x)) d x$

Step 4

Let $u = \sin (x)$ . Then $d u = \cos (x) d x$ , so $\frac{1}{\cos (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = \sin (x)$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $\sin (x)$ .

$\frac{d}{d x} [\sin (x)]$

Step 4.1.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\cos (x)$

$\cos (x)$

Step 4.2

Substitute the lower limit in for $x$ in $u = \sin (x)$ .

$u_{lower} = \sin (0)$

Step 4.3

The exact value of $\sin (0)$ is $0$ .

$u_{lower} = 0$

Step 4.4

Substitute the upper limit in for $x$ in $u = \sin (x)$ .

$u_{upper} = \sin (\frac{π}{2})$

Step 4.5

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$u_{upper} = 1$

Step 4.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 1$

Step 4.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{0}^{1} u^{7} {(1 - u^{2})}^{2} d u$

$\int_{0}^{1} u^{7} {(1 - u^{2})}^{2} d u$

Step 5

Expand $u^{7} {(1 - u^{2})}^{2}$ .

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Step 5.1

Rewrite ${(1 - u^{2})}^{2}$ as $(1 - u^{2}) (1 - u^{2})$ .

$\int_{0}^{1} u^{7} ((1 - u^{2}) (1 - u^{2})) d u$

Step 5.2

Apply the distributive property.

$\int_{0}^{1} u^{7} (1 (1 - u^{2}) - u^{2} (1 - u^{2})) d u$

Step 5.3

Apply the distributive property.

$\int_{0}^{1} u^{7} (1 \cdot 1 + 1 (- u^{2}) - u^{2} (1 - u^{2})) d u$

Step 5.4

Apply the distributive property.

$\int_{0}^{1} u^{7} (1 \cdot 1 + 1 (- u^{2}) - u^{2} \cdot 1 - u^{2} (- u^{2})) d u$

Step 5.5

Apply the distributive property.

$\int_{0}^{1} u^{7} (1 \cdot 1 + 1 (- u^{2})) + u^{7} (- u^{2} \cdot 1 - u^{2} (- u^{2})) d u$

Step 5.6

Apply the distributive property.

$\int_{0}^{1} u^{7} (1 \cdot 1) + u^{7} (1 (- u^{2})) + u^{7} (- u^{2} \cdot 1 - u^{2} (- u^{2})) d u$

Step 5.7

Apply the distributive property.

$\int_{0}^{1} u^{7} (1 \cdot 1) + u^{7} (1 (- u^{2})) + u^{7} (- u^{2} \cdot 1) + u^{7} (- u^{2} (- u^{2})) d u$

Step 5.8

Move $u^{7}$ .

$\int_{0}^{1} 1 \cdot 1 u^{7} + u^{7} (1 (- u^{2})) + u^{7} (- u^{2} \cdot 1) + u^{7} (- u^{2} (- u^{2})) d u$

Step 5.9

Move parentheses.

$\int_{0}^{1} 1 \cdot 1 u^{7} + u^{7} (1 \cdot - 1) u^{2} + u^{7} (- u^{2} \cdot 1) + u^{7} (- u^{2} (- u^{2})) d u$

Step 5.10

Move $u^{7}$ .

$\int_{0}^{1} 1 \cdot 1 u^{7} + 1 \cdot - 1 u^{7} u^{2} + u^{7} (- u^{2} \cdot 1) + u^{7} (- u^{2} (- u^{2})) d u$

Step 5.11

Move $u^{2}$ .

$\int_{0}^{1} 1 \cdot 1 u^{7} + 1 \cdot - 1 u^{7} u^{2} + u^{7} (- 1 \cdot 1 u^{2}) + u^{7} (- u^{2} (- u^{2})) d u$

Step 5.12

Move parentheses.

$\int_{0}^{1} 1 \cdot 1 u^{7} + 1 \cdot - 1 u^{7} u^{2} + u^{7} (- 1 \cdot 1) u^{2} + u^{7} (- u^{2} (- u^{2})) d u$

Step 5.13

Move $u^{7}$ .

$\int_{0}^{1} 1 \cdot 1 u^{7} + 1 \cdot - 1 u^{7} u^{2} - 1 \cdot 1 u^{7} u^{2} + u^{7} (- u^{2} (- u^{2})) d u$

Step 5.14

Move $u^{2}$ .

$\int_{0}^{1} 1 \cdot 1 u^{7} + 1 \cdot - 1 u^{7} u^{2} - 1 \cdot 1 u^{7} u^{2} + u^{7} (- 1 \cdot - 1 u^{2} u^{2}) d u$

Step 5.15

Move parentheses.

$\int_{0}^{1} 1 \cdot 1 u^{7} + 1 \cdot - 1 u^{7} u^{2} - 1 \cdot 1 u^{7} u^{2} + u^{7} (- 1 \cdot - 1 u^{2}) u^{2} d u$

Step 5.16

Move parentheses.

$\int_{0}^{1} 1 \cdot 1 u^{7} + 1 \cdot - 1 u^{7} u^{2} - 1 \cdot 1 u^{7} u^{2} + u^{7} (- 1 \cdot - 1) u^{2} u^{2} d u$

Step 5.17

Move $u^{7}$ .

$\int_{0}^{1} 1 \cdot 1 u^{7} + 1 \cdot - 1 u^{7} u^{2} - 1 \cdot 1 u^{7} u^{2} - 1 \cdot - 1 u^{7} u^{2} u^{2} d u$

Step 5.18

Multiply $1$ by $1$ .

$\int_{0}^{1} 1 u^{7} + 1 \cdot - 1 u^{7} u^{2} - 1 \cdot 1 u^{7} u^{2} - 1 \cdot - 1 u^{7} u^{2} u^{2} d u$

Step 5.19

Multiply $u^{7}$ by $1$ .

$\int_{0}^{1} u^{7} + 1 \cdot - 1 u^{7} u^{2} - 1 \cdot 1 u^{7} u^{2} - 1 \cdot - 1 u^{7} u^{2} u^{2} d u$

Step 5.20

Multiply $- 1$ by $1$ .

$\int_{0}^{1} u^{7} - u^{7} u^{2} - 1 \cdot 1 u^{7} u^{2} - 1 \cdot - 1 u^{7} u^{2} u^{2} d u$

Step 5.21

Factor out negative.

$\int_{0}^{1} u^{7} - (u^{7} u^{2}) - 1 \cdot 1 u^{7} u^{2} - 1 \cdot - 1 u^{7} u^{2} u^{2} d u$

Step 5.22

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{0}^{1} u^{7} - u^{7 + 2} - 1 \cdot 1 u^{7} u^{2} - 1 \cdot - 1 u^{7} u^{2} u^{2} d u$

Step 5.23

Add $7$ and $2$ .

$\int_{0}^{1} u^{7} - u^{9} - 1 \cdot 1 u^{7} u^{2} - 1 \cdot - 1 u^{7} u^{2} u^{2} d u$

Step 5.24

Multiply $- 1$ by $1$ .

$\int_{0}^{1} u^{7} - u^{9} - u^{7} u^{2} - 1 \cdot - 1 u^{7} u^{2} u^{2} d u$

Step 5.25

Factor out negative.

$\int_{0}^{1} u^{7} - u^{9} - (u^{7} u^{2}) - 1 \cdot - 1 u^{7} u^{2} u^{2} d u$

Step 5.26

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{0}^{1} u^{7} - u^{9} - u^{7 + 2} - 1 \cdot - 1 u^{7} u^{2} u^{2} d u$

Step 5.27

Add $7$ and $2$ .

$\int_{0}^{1} u^{7} - u^{9} - u^{9} - 1 \cdot - 1 u^{7} u^{2} u^{2} d u$

Step 5.28

Multiply $- 1$ by $- 1$ .

$\int_{0}^{1} u^{7} - u^{9} - u^{9} + 1 u^{7} u^{2} u^{2} d u$

Step 5.29

Multiply $u^{7}$ by $1$ .

$\int_{0}^{1} u^{7} - u^{9} - u^{9} + u^{7} u^{2} u^{2} d u$

Step 5.30

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{0}^{1} u^{7} - u^{9} - u^{9} + u^{7 + 2} u^{2} d u$

Step 5.31

Add $7$ and $2$ .

$\int_{0}^{1} u^{7} - u^{9} - u^{9} + u^{9} u^{2} d u$

Step 5.32

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{0}^{1} u^{7} - u^{9} - u^{9} + u^{9 + 2} d u$

Step 5.33

Add $9$ and $2$ .

$\int_{0}^{1} u^{7} - u^{9} - u^{9} + u^{11} d u$

Step 5.34

Subtract $u^{9}$ from $- u^{9}$ .

$\int_{0}^{1} u^{7} - 2 u^{9} + u^{11} d u$

Step 5.35

Reorder $- 2 u^{9}$ and $u^{11}$ .

$\int_{0}^{1} u^{7} + u^{11} - 2 u^{9} d u$

Step 5.36

Move $u^{7}$ .

$\int_{0}^{1} u^{11} - 2 u^{9} + u^{7} d u$

$\int_{0}^{1} u^{11} - 2 u^{9} + u^{7} d u$

Step 6

Split the single integral into multiple integrals.

$\int_{0}^{1} u^{11} d u + \int_{0}^{1} - 2 u^{9} d u + \int_{0}^{1} u^{7} d u$

Step 7

By the Power Rule, the integral of $u^{11}$ with respect to $u$ is $\frac{1}{12} u^{12}$ .

$\frac{1}{12} u^{12}]_{0}^{1} + \int_{0}^{1} - 2 u^{9} d u + \int_{0}^{1} u^{7} d u$

Step 8

Since $- 2$ is constant with respect to $u$ , move $- 2$ out of the integral.

$\frac{1}{12} u^{12}]_{0}^{1} - 2 \int_{0}^{1} u^{9} d u + \int_{0}^{1} u^{7} d u$

Step 9

By the Power Rule, the integral of $u^{9}$ with respect to $u$ is $\frac{1}{10} u^{10}$ .

$\frac{1}{12} u^{12}]_{0}^{1} - 2 (\frac{1}{10} u^{10}]_{0}^{1}) + \int_{0}^{1} u^{7} d u$

Step 10

Combine $\frac{1}{10}$ and $u^{10}$ .

$\frac{1}{12} u^{12}]_{0}^{1} - 2 (\frac{u^{10}}{10}]_{0}^{1}) + \int_{0}^{1} u^{7} d u$

Step 11

By the Power Rule, the integral of $u^{7}$ with respect to $u$ is $\frac{1}{8} u^{8}$ .

$\frac{1}{12} u^{12}]_{0}^{1} - 2 (\frac{u^{10}}{10}]_{0}^{1}) + \frac{1}{8} u^{8}]_{0}^{1}$

Step 12

Combine $\frac{1}{12} u^{12}]_{0}^{1}$ and $\frac{1}{8} u^{8}]_{0}^{1}$ .

$\frac{1}{12} u^{12} + \frac{1}{8} u^{8}]_{0}^{1} - 2 (\frac{u^{10}}{10}]_{0}^{1})$

Step 13

Substitute and simplify.

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Step 13.1

Evaluate $\frac{1}{12} u^{12} + \frac{1}{8} u^{8}$ at $1$ and at $0$ .

$(\frac{1}{12} \cdot 1^{12} + \frac{1}{8} \cdot 1^{8}) - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 (\frac{u^{10}}{10}]_{0}^{1})$

Step 13.2

Evaluate $\frac{u^{10}}{10}$ at $1$ and at $0$ .

$(\frac{1}{12} \cdot 1^{12} + \frac{1}{8} \cdot 1^{8}) - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3

Simplify.

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Step 13.3.1

One to any power is one.

$\frac{1}{12} \cdot 1 + \frac{1}{8} \cdot 1^{8} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.2

Multiply $\frac{1}{12}$ by $1$ .

$\frac{1}{12} + \frac{1}{8} \cdot 1^{8} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.3

One to any power is one.

$\frac{1}{12} + \frac{1}{8} \cdot 1 - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.4

Multiply $\frac{1}{8}$ by $1$ .

$\frac{1}{12} + \frac{1}{8} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.5

To write $\frac{1}{12}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{12} \cdot \frac{2}{2} + \frac{1}{8} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.6

To write $\frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{12} \cdot \frac{2}{2} + \frac{1}{8} \cdot \frac{3}{3} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.7

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.3.7.1

Multiply $\frac{1}{12}$ by $\frac{2}{2}$ .

$\frac{2}{12 \cdot 2} + \frac{1}{8} \cdot \frac{3}{3} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.7.2

Multiply $12$ by $2$ .

$\frac{2}{24} + \frac{1}{8} \cdot \frac{3}{3} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.7.3

Multiply $\frac{1}{8}$ by $\frac{3}{3}$ .

$\frac{2}{24} + \frac{3}{8 \cdot 3} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.7.4

Multiply $8$ by $3$ .

$\frac{2}{24} + \frac{3}{24} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

$\frac{2}{24} + \frac{3}{24} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.8

Combine the numerators over the common denominator.

$\frac{2 + 3}{24} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.9

Add $2$ and $3$ .

$\frac{5}{24} - (\frac{1}{12} \cdot 0^{12} + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.10

Raising $0$ to any positive power yields $0$ .

$\frac{5}{24} - (\frac{1}{12} \cdot 0 + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.11

Multiply $\frac{1}{12}$ by $0$ .

$\frac{5}{24} - (0 + \frac{1}{8} \cdot 0^{8}) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.12

Raising $0$ to any positive power yields $0$ .

$\frac{5}{24} - (0 + \frac{1}{8} \cdot 0) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.13

Multiply $\frac{1}{8}$ by $0$ .

$\frac{5}{24} - (0 + 0) - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.14

Add $0$ and $0$ .

$\frac{5}{24} - 0 - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.15

Multiply $- 1$ by $0$ .

$\frac{5}{24} + 0 - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.16

Add $\frac{5}{24}$ and $0$ .

$\frac{5}{24} - 2 ((\frac{1^{10}}{10}) - \frac{0^{10}}{10})$

Step 13.3.17

One to any power is one.

$\frac{5}{24} - 2 (\frac{1}{10} - \frac{0^{10}}{10})$

Step 13.3.18

Raising $0$ to any positive power yields $0$ .

$\frac{5}{24} - 2 (\frac{1}{10} - \frac{0}{10})$

Step 13.3.19

Cancel the common factor of $0$ and $10$ .

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Step 13.3.19.1

Factor $10$ out of $0$ .

$\frac{5}{24} - 2 (\frac{1}{10} - \frac{10 (0)}{10})$

Step 13.3.19.2

Cancel the common factors.

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Step 13.3.19.2.1

Factor $10$ out of $10$ .

$\frac{5}{24} - 2 (\frac{1}{10} - \frac{10 \cdot 0}{10 \cdot 1})$

Step 13.3.19.2.2

Cancel the common factor.

$\frac{5}{24} - 2 (\frac{1}{10} - \frac{10 \cdot 0}{10 \cdot 1})$

Step 13.3.19.2.3

Rewrite the expression.

$\frac{5}{24} - 2 (\frac{1}{10} - \frac{0}{1})$

Step 13.3.19.2.4

Divide $0$ by $1$ .

$\frac{5}{24} - 2 (\frac{1}{10} - 0)$

$\frac{5}{24} - 2 (\frac{1}{10} - 0)$

$\frac{5}{24} - 2 (\frac{1}{10} - 0)$

Step 13.3.20

Multiply $- 1$ by $0$ .

$\frac{5}{24} - 2 (\frac{1}{10} + 0)$

Step 13.3.21

Add $\frac{1}{10}$ and $0$ .

$\frac{5}{24} - 2 (\frac{1}{10})$

Step 13.3.22

Combine $- 2$ and $\frac{1}{10}$ .

$\frac{5}{24} + \frac{- 2}{10}$

Step 13.3.23

Cancel the common factor of $- 2$ and $10$ .

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Step 13.3.23.1

Factor $2$ out of $- 2$ .

$\frac{5}{24} + \frac{2 (- 1)}{10}$

Step 13.3.23.2

Cancel the common factors.

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Step 13.3.23.2.1

Factor $2$ out of $10$ .

$\frac{5}{24} + \frac{2 \cdot - 1}{2 \cdot 5}$

Step 13.3.23.2.2

Cancel the common factor.

$\frac{5}{24} + \frac{2 \cdot - 1}{2 \cdot 5}$

Step 13.3.23.2.3

Rewrite the expression.

$\frac{5}{24} + \frac{- 1}{5}$

$\frac{5}{24} + \frac{- 1}{5}$

$\frac{5}{24} + \frac{- 1}{5}$

Step 13.3.24

Move the negative in front of the fraction.

$\frac{5}{24} - \frac{1}{5}$

Step 13.3.25

To write $\frac{5}{24}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{5}{24} \cdot \frac{5}{5} - \frac{1}{5}$

Step 13.3.26

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{24}{24}$ .

$\frac{5}{24} \cdot \frac{5}{5} - \frac{1}{5} \cdot \frac{24}{24}$

Step 13.3.27

Write each expression with a common denominator of $120$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.3.27.1

Multiply $\frac{5}{24}$ by $\frac{5}{5}$ .

$\frac{5 \cdot 5}{24 \cdot 5} - \frac{1}{5} \cdot \frac{24}{24}$

Step 13.3.27.2

Multiply $24$ by $5$ .

$\frac{5 \cdot 5}{120} - \frac{1}{5} \cdot \frac{24}{24}$

Step 13.3.27.3

Multiply $\frac{1}{5}$ by $\frac{24}{24}$ .

$\frac{5 \cdot 5}{120} - \frac{24}{5 \cdot 24}$

Step 13.3.27.4

Multiply $5$ by $24$ .

$\frac{5 \cdot 5}{120} - \frac{24}{120}$

$\frac{5 \cdot 5}{120} - \frac{24}{120}$

Step 13.3.28

Combine the numerators over the common denominator.

$\frac{5 \cdot 5 - 24}{120}$

Step 13.3.29

Simplify the numerator.

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Step 13.3.29.1

Multiply $5$ by $5$ .

$\frac{25 - 24}{120}$

Step 13.3.29.2

Subtract $24$ from $25$ .

$\frac{1}{120}$

$\frac{1}{120}$

$\frac{1}{120}$

$\frac{1}{120}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{120}$

Decimal Form:

$0.008\overline{3}$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$